I know there aren't that many, but at least there's a few. I'm not including games that take place during the calendar year either in real time (like Animal Crossing) or not (like Persona).
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Christmas NiGHTS into Dreams: pretty self-explanatory, it's also included in the later console ports, although it's missing the cool presents you could get, including being able to play as
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James Pond 2: Codename: RoboCod: takes place in Santa's castle and you have to save Christmas.
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Yakuza: surprisingly many of them take place at Christmas.
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Daze Before Christmas: a platformer for the Mega Drive and SNES that's not particularly good.
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Xmas Lemmings: basically demos of Lemmings with Christmas-themed levels, there were a bunch of these for PC and Amiga.
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Tyrian: Originally a DOS shmup that was released as freeware and now has open source ports, there's a Christmas mode that activates if the system clock is set to December. I'm pretty sure there were other DOS games with Christmas modes too.
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Duke: Nuclear Winter: a Christmas-themed expansion for Duke Nukem 3D.
It's mostly inconsequential to the plot, but there's a part involving a Santa Claus and a version of Jingle Bells that plays.
I never played Origins, but I remember one of the Batman Arkham games being Christmasy. Was there another that had a snowy environment?
Arkham City has a winter setting, just not Christmas-related.
Doesn't it also have a Batman/Joker Pieta-ish image somewhere towards the end? I can see where that all blurs together as "Christmas" in my half-remembered memories.
I found a YouTube link in your comment. Here are links to the same video on alternative frontends that protect your privacy: